direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C3×C4.12D20, C12.64D20, C60.218D4, C4.12(C3×D20), C20.47(C3×D4), (C6×Dic5).2C4, C22.5(D5×C12), C4.Dic5.3C6, (C2×C12).212D10, C15⋊8(C4.10D4), (C2×Dic10).7C6, (C2×Dic5).1C12, (C5×M4(2)).2C6, M4(2).2(C3×D5), (C3×M4(2)).2D5, C12.115(C5⋊D4), C30.88(C22⋊C4), (C2×C60).277C22, (C6×Dic10).18C2, (C15×M4(2)).2C2, C6.41(D10⋊C4), (C2×C4).2(C6×D5), (C2×C6).39(C4×D5), C4.22(C3×C5⋊D4), C5⋊2(C3×C4.10D4), (C2×C20).13(C2×C6), (C2×C30).120(C2×C4), (C2×C10).23(C2×C12), C10.20(C3×C22⋊C4), (C3×C4.Dic5).7C2, C2.10(C3×D10⋊C4), SmallGroup(480,102)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4.12D20
G = < a,b,c,d | a3=b20=1, c4=d2=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b5c3 >
Subgroups: 224 in 76 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, C10, C10, C12, C12, C2×C6, C15, M4(2), M4(2), C2×Q8, Dic5, C20, C2×C10, C24, C2×C12, C2×C12, C3×Q8, C30, C30, C4.10D4, C5⋊2C8, C40, Dic10, C2×Dic5, C2×C20, C3×M4(2), C3×M4(2), C6×Q8, C3×Dic5, C60, C2×C30, C4.Dic5, C5×M4(2), C2×Dic10, C3×C4.10D4, C3×C5⋊2C8, C120, C3×Dic10, C6×Dic5, C2×C60, C4.12D20, C3×C4.Dic5, C15×M4(2), C6×Dic10, C3×C4.12D20
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, D5, C12, C2×C6, C22⋊C4, D10, C2×C12, C3×D4, C3×D5, C4.10D4, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C6×D5, D10⋊C4, C3×C4.10D4, D5×C12, C3×D20, C3×C5⋊D4, C4.12D20, C3×D10⋊C4, C3×C4.12D20
►On 240 points
Generators in S240
(1 53 33)(2 54 34)(3 55 35)(4 56 36)(5 57 37)(6 58 38)(7 59 39)(8 60 40)(9 41 21)(10 42 22)(11 43 23)(12 44 24)(13 45 25)(14 46 26)(15 47 27)(16 48 28)(17 49 29)(18 50 30)(19 51 31)(20 52 32)(61 112 81)(62 113 82)(63 114 83)(64 115 84)(65 116 85)(66 117 86)(67 118 87)(68 119 88)(69 120 89)(70 101 90)(71 102 91)(72 103 92)(73 104 93)(74 105 94)(75 106 95)(76 107 96)(77 108 97)(78 109 98)(79 110 99)(80 111 100)(121 161 141)(122 162 142)(123 163 143)(124 164 144)(125 165 145)(126 166 146)(127 167 147)(128 168 148)(129 169 149)(130 170 150)(131 171 151)(132 172 152)(133 173 153)(134 174 154)(135 175 155)(136 176 156)(137 177 157)(138 178 158)(139 179 159)(140 180 160)(181 221 211)(182 222 212)(183 223 213)(184 224 214)(185 225 215)(186 226 216)(187 227 217)(188 228 218)(189 229 219)(190 230 220)(191 231 201)(192 232 202)(193 233 203)(194 234 204)(195 235 205)(196 236 206)(197 237 207)(198 238 208)(199 239 209)(200 240 210)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220)(221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 196 6 191 11 186 16 181)(2 195 7 190 12 185 17 200)(3 194 8 189 13 184 18 199)(4 193 9 188 14 183 19 198)(5 192 10 187 15 182 20 197)(21 218 26 213 31 208 36 203)(22 217 27 212 32 207 37 202)(23 216 28 211 33 206 38 201)(24 215 29 210 34 205 39 220)(25 214 30 209 35 204 40 219)(41 228 46 223 51 238 56 233)(42 227 47 222 52 237 57 232)(43 226 48 221 53 236 58 231)(44 225 49 240 54 235 59 230)(45 224 50 239 55 234 60 229)(61 135 76 140 71 125 66 130)(62 134 77 139 72 124 67 129)(63 133 78 138 73 123 68 128)(64 132 79 137 74 122 69 127)(65 131 80 136 75 121 70 126)(81 155 96 160 91 145 86 150)(82 154 97 159 92 144 87 149)(83 153 98 158 93 143 88 148)(84 152 99 157 94 142 89 147)(85 151 100 156 95 141 90 146)(101 166 116 171 111 176 106 161)(102 165 117 170 112 175 107 180)(103 164 118 169 113 174 108 179)(104 163 119 168 114 173 109 178)(105 162 120 167 115 172 110 177)
(1 126 11 136)(2 125 12 135)(3 124 13 134)(4 123 14 133)(5 122 15 132)(6 121 16 131)(7 140 17 130)(8 139 18 129)(9 138 19 128)(10 137 20 127)(21 158 31 148)(22 157 32 147)(23 156 33 146)(24 155 34 145)(25 154 35 144)(26 153 36 143)(27 152 37 142)(28 151 38 141)(29 150 39 160)(30 149 40 159)(41 178 51 168)(42 177 52 167)(43 176 53 166)(44 175 54 165)(45 174 55 164)(46 173 56 163)(47 172 57 162)(48 171 58 161)(49 170 59 180)(50 169 60 179)(61 200 71 190)(62 199 72 189)(63 198 73 188)(64 197 74 187)(65 196 75 186)(66 195 76 185)(67 194 77 184)(68 193 78 183)(69 192 79 182)(70 191 80 181)(81 210 91 220)(82 209 92 219)(83 208 93 218)(84 207 94 217)(85 206 95 216)(86 205 96 215)(87 204 97 214)(88 203 98 213)(89 202 99 212)(90 201 100 211)(101 231 111 221)(102 230 112 240)(103 229 113 239)(104 228 114 238)(105 227 115 237)(106 226 116 236)(107 225 117 235)(108 224 118 234)(109 223 119 233)(110 222 120 232)
G:=sub<Sym(240)| (1,53,33)(2,54,34)(3,55,35)(4,56,36)(5,57,37)(6,58,38)(7,59,39)(8,60,40)(9,41,21)(10,42,22)(11,43,23)(12,44,24)(13,45,25)(14,46,26)(15,47,27)(16,48,28)(17,49,29)(18,50,30)(19,51,31)(20,52,32)(61,112,81)(62,113,82)(63,114,83)(64,115,84)(65,116,85)(66,117,86)(67,118,87)(68,119,88)(69,120,89)(70,101,90)(71,102,91)(72,103,92)(73,104,93)(74,105,94)(75,106,95)(76,107,96)(77,108,97)(78,109,98)(79,110,99)(80,111,100)(121,161,141)(122,162,142)(123,163,143)(124,164,144)(125,165,145)(126,166,146)(127,167,147)(128,168,148)(129,169,149)(130,170,150)(131,171,151)(132,172,152)(133,173,153)(134,174,154)(135,175,155)(136,176,156)(137,177,157)(138,178,158)(139,179,159)(140,180,160)(181,221,211)(182,222,212)(183,223,213)(184,224,214)(185,225,215)(186,226,216)(187,227,217)(188,228,218)(189,229,219)(190,230,220)(191,231,201)(192,232,202)(193,233,203)(194,234,204)(195,235,205)(196,236,206)(197,237,207)(198,238,208)(199,239,209)(200,240,210), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,196,6,191,11,186,16,181)(2,195,7,190,12,185,17,200)(3,194,8,189,13,184,18,199)(4,193,9,188,14,183,19,198)(5,192,10,187,15,182,20,197)(21,218,26,213,31,208,36,203)(22,217,27,212,32,207,37,202)(23,216,28,211,33,206,38,201)(24,215,29,210,34,205,39,220)(25,214,30,209,35,204,40,219)(41,228,46,223,51,238,56,233)(42,227,47,222,52,237,57,232)(43,226,48,221,53,236,58,231)(44,225,49,240,54,235,59,230)(45,224,50,239,55,234,60,229)(61,135,76,140,71,125,66,130)(62,134,77,139,72,124,67,129)(63,133,78,138,73,123,68,128)(64,132,79,137,74,122,69,127)(65,131,80,136,75,121,70,126)(81,155,96,160,91,145,86,150)(82,154,97,159,92,144,87,149)(83,153,98,158,93,143,88,148)(84,152,99,157,94,142,89,147)(85,151,100,156,95,141,90,146)(101,166,116,171,111,176,106,161)(102,165,117,170,112,175,107,180)(103,164,118,169,113,174,108,179)(104,163,119,168,114,173,109,178)(105,162,120,167,115,172,110,177), (1,126,11,136)(2,125,12,135)(3,124,13,134)(4,123,14,133)(5,122,15,132)(6,121,16,131)(7,140,17,130)(8,139,18,129)(9,138,19,128)(10,137,20,127)(21,158,31,148)(22,157,32,147)(23,156,33,146)(24,155,34,145)(25,154,35,144)(26,153,36,143)(27,152,37,142)(28,151,38,141)(29,150,39,160)(30,149,40,159)(41,178,51,168)(42,177,52,167)(43,176,53,166)(44,175,54,165)(45,174,55,164)(46,173,56,163)(47,172,57,162)(48,171,58,161)(49,170,59,180)(50,169,60,179)(61,200,71,190)(62,199,72,189)(63,198,73,188)(64,197,74,187)(65,196,75,186)(66,195,76,185)(67,194,77,184)(68,193,78,183)(69,192,79,182)(70,191,80,181)(81,210,91,220)(82,209,92,219)(83,208,93,218)(84,207,94,217)(85,206,95,216)(86,205,96,215)(87,204,97,214)(88,203,98,213)(89,202,99,212)(90,201,100,211)(101,231,111,221)(102,230,112,240)(103,229,113,239)(104,228,114,238)(105,227,115,237)(106,226,116,236)(107,225,117,235)(108,224,118,234)(109,223,119,233)(110,222,120,232)>;
G:=Group( (1,53,33)(2,54,34)(3,55,35)(4,56,36)(5,57,37)(6,58,38)(7,59,39)(8,60,40)(9,41,21)(10,42,22)(11,43,23)(12,44,24)(13,45,25)(14,46,26)(15,47,27)(16,48,28)(17,49,29)(18,50,30)(19,51,31)(20,52,32)(61,112,81)(62,113,82)(63,114,83)(64,115,84)(65,116,85)(66,117,86)(67,118,87)(68,119,88)(69,120,89)(70,101,90)(71,102,91)(72,103,92)(73,104,93)(74,105,94)(75,106,95)(76,107,96)(77,108,97)(78,109,98)(79,110,99)(80,111,100)(121,161,141)(122,162,142)(123,163,143)(124,164,144)(125,165,145)(126,166,146)(127,167,147)(128,168,148)(129,169,149)(130,170,150)(131,171,151)(132,172,152)(133,173,153)(134,174,154)(135,175,155)(136,176,156)(137,177,157)(138,178,158)(139,179,159)(140,180,160)(181,221,211)(182,222,212)(183,223,213)(184,224,214)(185,225,215)(186,226,216)(187,227,217)(188,228,218)(189,229,219)(190,230,220)(191,231,201)(192,232,202)(193,233,203)(194,234,204)(195,235,205)(196,236,206)(197,237,207)(198,238,208)(199,239,209)(200,240,210), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,196,6,191,11,186,16,181)(2,195,7,190,12,185,17,200)(3,194,8,189,13,184,18,199)(4,193,9,188,14,183,19,198)(5,192,10,187,15,182,20,197)(21,218,26,213,31,208,36,203)(22,217,27,212,32,207,37,202)(23,216,28,211,33,206,38,201)(24,215,29,210,34,205,39,220)(25,214,30,209,35,204,40,219)(41,228,46,223,51,238,56,233)(42,227,47,222,52,237,57,232)(43,226,48,221,53,236,58,231)(44,225,49,240,54,235,59,230)(45,224,50,239,55,234,60,229)(61,135,76,140,71,125,66,130)(62,134,77,139,72,124,67,129)(63,133,78,138,73,123,68,128)(64,132,79,137,74,122,69,127)(65,131,80,136,75,121,70,126)(81,155,96,160,91,145,86,150)(82,154,97,159,92,144,87,149)(83,153,98,158,93,143,88,148)(84,152,99,157,94,142,89,147)(85,151,100,156,95,141,90,146)(101,166,116,171,111,176,106,161)(102,165,117,170,112,175,107,180)(103,164,118,169,113,174,108,179)(104,163,119,168,114,173,109,178)(105,162,120,167,115,172,110,177), (1,126,11,136)(2,125,12,135)(3,124,13,134)(4,123,14,133)(5,122,15,132)(6,121,16,131)(7,140,17,130)(8,139,18,129)(9,138,19,128)(10,137,20,127)(21,158,31,148)(22,157,32,147)(23,156,33,146)(24,155,34,145)(25,154,35,144)(26,153,36,143)(27,152,37,142)(28,151,38,141)(29,150,39,160)(30,149,40,159)(41,178,51,168)(42,177,52,167)(43,176,53,166)(44,175,54,165)(45,174,55,164)(46,173,56,163)(47,172,57,162)(48,171,58,161)(49,170,59,180)(50,169,60,179)(61,200,71,190)(62,199,72,189)(63,198,73,188)(64,197,74,187)(65,196,75,186)(66,195,76,185)(67,194,77,184)(68,193,78,183)(69,192,79,182)(70,191,80,181)(81,210,91,220)(82,209,92,219)(83,208,93,218)(84,207,94,217)(85,206,95,216)(86,205,96,215)(87,204,97,214)(88,203,98,213)(89,202,99,212)(90,201,100,211)(101,231,111,221)(102,230,112,240)(103,229,113,239)(104,228,114,238)(105,227,115,237)(106,226,116,236)(107,225,117,235)(108,224,118,234)(109,223,119,233)(110,222,120,232) );
G=PermutationGroup([[(1,53,33),(2,54,34),(3,55,35),(4,56,36),(5,57,37),(6,58,38),(7,59,39),(8,60,40),(9,41,21),(10,42,22),(11,43,23),(12,44,24),(13,45,25),(14,46,26),(15,47,27),(16,48,28),(17,49,29),(18,50,30),(19,51,31),(20,52,32),(61,112,81),(62,113,82),(63,114,83),(64,115,84),(65,116,85),(66,117,86),(67,118,87),(68,119,88),(69,120,89),(70,101,90),(71,102,91),(72,103,92),(73,104,93),(74,105,94),(75,106,95),(76,107,96),(77,108,97),(78,109,98),(79,110,99),(80,111,100),(121,161,141),(122,162,142),(123,163,143),(124,164,144),(125,165,145),(126,166,146),(127,167,147),(128,168,148),(129,169,149),(130,170,150),(131,171,151),(132,172,152),(133,173,153),(134,174,154),(135,175,155),(136,176,156),(137,177,157),(138,178,158),(139,179,159),(140,180,160),(181,221,211),(182,222,212),(183,223,213),(184,224,214),(185,225,215),(186,226,216),(187,227,217),(188,228,218),(189,229,219),(190,230,220),(191,231,201),(192,232,202),(193,233,203),(194,234,204),(195,235,205),(196,236,206),(197,237,207),(198,238,208),(199,239,209),(200,240,210)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220),(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,196,6,191,11,186,16,181),(2,195,7,190,12,185,17,200),(3,194,8,189,13,184,18,199),(4,193,9,188,14,183,19,198),(5,192,10,187,15,182,20,197),(21,218,26,213,31,208,36,203),(22,217,27,212,32,207,37,202),(23,216,28,211,33,206,38,201),(24,215,29,210,34,205,39,220),(25,214,30,209,35,204,40,219),(41,228,46,223,51,238,56,233),(42,227,47,222,52,237,57,232),(43,226,48,221,53,236,58,231),(44,225,49,240,54,235,59,230),(45,224,50,239,55,234,60,229),(61,135,76,140,71,125,66,130),(62,134,77,139,72,124,67,129),(63,133,78,138,73,123,68,128),(64,132,79,137,74,122,69,127),(65,131,80,136,75,121,70,126),(81,155,96,160,91,145,86,150),(82,154,97,159,92,144,87,149),(83,153,98,158,93,143,88,148),(84,152,99,157,94,142,89,147),(85,151,100,156,95,141,90,146),(101,166,116,171,111,176,106,161),(102,165,117,170,112,175,107,180),(103,164,118,169,113,174,108,179),(104,163,119,168,114,173,109,178),(105,162,120,167,115,172,110,177)], [(1,126,11,136),(2,125,12,135),(3,124,13,134),(4,123,14,133),(5,122,15,132),(6,121,16,131),(7,140,17,130),(8,139,18,129),(9,138,19,128),(10,137,20,127),(21,158,31,148),(22,157,32,147),(23,156,33,146),(24,155,34,145),(25,154,35,144),(26,153,36,143),(27,152,37,142),(28,151,38,141),(29,150,39,160),(30,149,40,159),(41,178,51,168),(42,177,52,167),(43,176,53,166),(44,175,54,165),(45,174,55,164),(46,173,56,163),(47,172,57,162),(48,171,58,161),(49,170,59,180),(50,169,60,179),(61,200,71,190),(62,199,72,189),(63,198,73,188),(64,197,74,187),(65,196,75,186),(66,195,76,185),(67,194,77,184),(68,193,78,183),(69,192,79,182),(70,191,80,181),(81,210,91,220),(82,209,92,219),(83,208,93,218),(84,207,94,217),(85,206,95,216),(86,205,96,215),(87,204,97,214),(88,203,98,213),(89,202,99,212),(90,201,100,211),(101,231,111,221),(102,230,112,240),(103,229,113,239),(104,228,114,238),(105,227,115,237),(106,226,116,236),(107,225,117,235),(108,224,118,234),(109,223,119,233),(110,222,120,232)]])
93 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 24E | 24F | 24G | 24H | 30A | 30B | 30C | 30D | 30E | 30F | 30G | 30H | 40A | ··· | 40H | 60A | ··· | 60H | 60I | 60J | 60K | 60L | 120A | ··· | 120P |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | ··· | 40 | 60 | ··· | 60 | 60 | 60 | 60 | 60 | 120 | ··· | 120 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 20 | 20 | 2 | 2 | 1 | 1 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
93 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | - | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D4 | D5 | D10 | C3×D4 | C3×D5 | D20 | C5⋊D4 | C4×D5 | C6×D5 | C3×D20 | C3×C5⋊D4 | D5×C12 | C4.10D4 | C3×C4.10D4 | C4.12D20 | C3×C4.12D20 |
| kernel | C3×C4.12D20 | C3×C4.Dic5 | C15×M4(2) | C6×Dic10 | C4.12D20 | C6×Dic5 | C4.Dic5 | C5×M4(2) | C2×Dic10 | C2×Dic5 | C60 | C3×M4(2) | C2×C12 | C20 | M4(2) | C12 | C12 | C2×C6 | C2×C4 | C4 | C4 | C22 | C15 | C5 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 2 | 4 | 8 |
Matrix representation of C3×C4.12D20 ►in GL4(𝔽241) generated by
| 15 | 0 | 0 | 0 |
| 0 | 15 | 0 | 0 |
| 0 | 0 | 15 | 0 |
| 0 | 0 | 0 | 15 |
| 238 | 197 | 0 | 0 |
| 44 | 163 | 0 | 0 |
| 0 | 0 | 41 | 197 |
| 0 | 0 | 163 | 119 |
| 60 | 148 | 196 | 203 |
| 75 | 181 | 225 | 83 |
| 47 | 36 | 135 | 181 |
| 70 | 230 | 119 | 106 |
| 60 | 148 | 0 | 0 |
| 75 | 181 | 0 | 0 |
| 0 | 0 | 106 | 60 |
| 0 | 0 | 122 | 135 |
G:=sub<GL(4,GF(241))| [15,0,0,0,0,15,0,0,0,0,15,0,0,0,0,15],[238,44,0,0,197,163,0,0,0,0,41,163,0,0,197,119],[60,75,47,70,148,181,36,230,196,225,135,119,203,83,181,106],[60,75,0,0,148,181,0,0,0,0,106,122,0,0,60,135] >;
C3×C4.12D20 in GAP, Magma, Sage, TeX
C_3\times C_4._{12}D_{20} % in TeX
G:=Group("C3xC4.12D20"); // GroupNames label
G:=SmallGroup(480,102);
// by ID
G=gap.SmallGroup(480,102);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,336,365,92,1683,136,1271,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^20=1,c^4=d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^5*c^3>;
// generators/relations